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Theorem cofu2nd 17147
Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b 𝐵 = (Base‘𝐶)
cofuval.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g (𝜑𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x (𝜑𝑋𝐵)
cofu2nd.y (𝜑𝑌𝐵)
Assertion
Ref Expression
cofu2nd (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))

Proof of Theorem cofu2nd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofuval.b . . . . 5 𝐵 = (Base‘𝐶)
2 cofuval.f . . . . 5 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
3 cofuval.g . . . . 5 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
41, 2, 3cofuval 17144 . . . 4 (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)
54fveq2d 6649 . . 3 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (2nd ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩))
6 fvex 6658 . . . . 5 (1st𝐺) ∈ V
7 fvex 6658 . . . . 5 (1st𝐹) ∈ V
86, 7coex 7617 . . . 4 ((1st𝐺) ∘ (1st𝐹)) ∈ V
91fvexi 6659 . . . . 5 𝐵 ∈ V
109, 9mpoex 7760 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))) ∈ V
118, 10op2nd 7680 . . 3 (2nd ‘⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
125, 11eqtrdi 2849 . 2 (𝜑 → (2nd ‘(𝐺func 𝐹)) = (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦))))
13 simprl 770 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
1413fveq2d 6649 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑋))
15 simprr 772 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
1615fveq2d 6649 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((1st𝐹)‘𝑦) = ((1st𝐹)‘𝑌))
1714, 16oveq12d 7153 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) = (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)))
1813, 15oveq12d 7153 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑌))
1917, 18coeq12d 5699 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
20 cofu2nd.x . 2 (𝜑𝑋𝐵)
21 cofu2nd.y . 2 (𝜑𝑌𝐵)
22 ovex 7168 . . . 4 (((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∈ V
23 ovex 7168 . . . 4 (𝑋(2nd𝐹)𝑌) ∈ V
2422, 23coex 7617 . . 3 ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)) ∈ V
2524a1i 11 . 2 (𝜑 → ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)) ∈ V)
2612, 19, 20, 21, 25ovmpod 7281 1 (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  cop 4531  ccom 5523  cfv 6324  (class class class)co 7135  cmpo 7137  1st c1st 7669  2nd c2nd 7670  Basecbs 16475   Func cfunc 17116  func ccofu 17118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-ixp 8445  df-func 17120  df-cofu 17122
This theorem is referenced by:  cofu2  17148  cofucl  17150  cofuass  17151  cofull  17196  cofth  17197  catciso  17359
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